Extending wavelet regularity beyond Gevrey classes

We construct a smooth orthonormal wavelet $\psi$ such that both $\psi$ and its Fourier transform $\widehat{\psi}$ belong to the extended Gevrey class $\mathcal{E}_{\sigma}(\mathbb{R})$ for $\sigma > 1$, providing an example that lies beyond all classical Gevrey classes. Our approach uses the idea of invariant cycles to extend the initial Lemari\'e-Meyer support of the low-pass filter $m_0$ from $[-\frac{2\pi}{3}, \frac{2\pi}{3}]$ to $[-\frac{4\pi}{5}, \frac{4\pi}{5}]$. This extension allows us to control the decay rate of $m_0$ near $\frac{2\pi}{3}$, which yields global decay estimates for $\psi$ and $\hat\psi$. In addition, the decay rates are described using special functions involving the Lambert W function, which plays an important role in our construction.